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 A322367 Number of disconnected or empty integer partitions of n. 3
 1, 0, 1, 2, 3, 6, 7, 14, 17, 27, 34, 54, 63, 98, 118, 165, 207, 287, 345, 474, 574, 757, 931, 1212, 1463, 1890, 2292, 2898, 3515, 4413, 5303 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS An integer partition is connected if the prime factorizations of its parts form a connected hypergraph. It is disconnected if it can be separated into two or more integer partitions with relatively prime products. For example, the integer partition (654321) has three connected components: (6432)(5)(1). LINKS EXAMPLE The a(3) = 2 through a(9) = 27 disconnected integer partitions:   (21)   (31)    (32)     (51)      (43)       (53)        (54)   (111)  (211)   (41)     (321)     (52)       (71)        (72)          (1111)  (221)    (411)     (61)       (332)       (81)                  (311)    (2211)    (322)      (431)       (432)                  (2111)   (3111)    (331)      (521)       (441)                  (11111)  (21111)   (421)      (611)       (522)                           (111111)  (511)      (3221)      (531)                                     (2221)     (3311)      (621)                                     (3211)     (4211)      (711)                                     (4111)     (5111)      (3222)                                     (22111)    (22211)     (3321)                                     (31111)    (32111)     (4221)                                     (211111)   (41111)     (4311)                                     (1111111)  (221111)    (5211)                                                (311111)    (6111)                                                (2111111)   (22221)                                                (11111111)  (32211)                                                            (33111)                                                            (42111)                                                            (51111)                                                            (222111)                                                            (321111)                                                            (411111)                                                            (2211111)                                                            (3111111)                                                            (21111111)                                                            (111111111) MATHEMATICA zsm[s_]:=With[{c=Select[Tuples[Range[Length[s]], 2], And[Less@@#, GCD@@s[[#]]]>1&]}, If[c=={}, s, zsm[Sort[Append[Delete[s, List/@c[[1]]], LCM@@s[[c[[1]]]]]]]]]; Table[Length[Select[IntegerPartitions[n], Length[zsm[#]]!=1&]], {n, 20}] CROSSREFS Cf. A054921, A218970, A286518, A322335, A304714, A304716, A305078, A305079, A322306, A322307, A322337, A322338, A322368, A322369. Sequence in context: A191615 A018606 A117087 * A319811 A000837 A200144 Adjacent sequences:  A322364 A322365 A322366 * A322368 A322369 A322370 KEYWORD nonn,more AUTHOR Gus Wiseman, Dec 04 2018 STATUS approved

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Last modified October 23 07:11 EDT 2019. Contains 328336 sequences. (Running on oeis4.)